FIG. 1 represents a simplified target tracking system 10. System 10 tracks a target, illustrated as being an aircraft 12, by the use of a radar system 14. Radar system 14 includes a radar antenna 14a, which transmits and receives radar signals illustrated by “lightning bolt” symbols 16. Portions of the transmitted signals 16 are reflected by target 12 and return to the radar antenna 14a. The returned signals allow the generation of measurements at an output port 14o of radar system 14. These measurements include values of at least target position, possibly in the form of range and angles from the radar system 14. The measurements are applied to a processing arrangement 16, which determines from the measurements various target parameters, which may include course (direction of motion), speed, and target type. The estimated position of the target, and possibly other information, is provided to a utilization apparatus or user, illustrated in this case as being a radar display 18. The operator (or possibly automated decision making equipment) can make decisions as to actions to be taken in response to the displayed information. It should be understood that the radar tracking system 10 of FIG. 1 is only one embodiment of a general class of estimation systems which may include controlling nuclear, chemical, or manufacturing factories or facilities, control processes subject to external parameter changes, attitude control of a space station subject to vibrations, traction control of an automobile subject to weather conditions, and the like.
Consider the problem of tracking an airplane whose trajectory in three dimensions is an arbitrary curve with bounded instantaneous turn rate and tangential acceleration. The parameters of this tracking problem are the turn rate ω (which can be related to the curvature of the trajectory) and the tangential acceleration α. These parameters, ω and α, are neither exclusively constant nor strictly white noise stochastic processes, but vary arbitrarily in time within physical bounds. Time dependent, but bounded, parameters ω and α typically represent a maneuvering target as described in Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory, Algorithms, and Software, New York, N.Y.: John Wiley & Sons, Inc., 2001, and in X. R. Li and V. P. Jilkov, “A Survey of Maneuvering Target Tracking—Part IV: Decision-Based Methods,” Proceedings of SPIE Vol. 4728 (2002), pp. 511–534.
This problem belongs to a more general problem of estimating the state of a system that depends on time dependent parameters with unknown values but with known bounds. In some situations, the Kalman filter solves this problem by including the parameters as part of an augmented state to be estimated, as described in C. Bembenek, T. A. Chmielewski, Jr., and P. R. Kalata, “Observability Conditions for Biased Linear Time Invariant Systems,” Proceedings of the American Control Conference, pp. 1180–1184, Philadelphia, Pa., June 1998, B. Friedland, “Treatment of Bias in Recursive Filtering,” IEEE Transactions on Automatic Control, pp. 359–367, Vol. AC-14, No. 4, August 1969, and D. Haessig and B. Friedland, “Separate-Bias Estimation with Reduced-Order Kalman Filters,” IEEE Transactions on Automatic Control, pp. 983–987, Vol. 43, No. 7, July 1998. Such a filter will be called a “full state” estimator. However, the parameters may vary too erratically to be considered as observables, as noted by G. J. Portmann, J. R. Moore, and W. G. Bath, “Separated Covariance Filtering,” Record of the IEEE 1990 International Radar Conference, 1990, pp. 456–460, or there may be too many parameters to estimate. In the case in which parameters cannot be estimated, filters, which do not augment the state vector with these parameters, often give better performance. Such a filter will be called a “reduced state” estimator. More generally, a “reduced state” or “reduced order” estimator uses fewer states than would be required to completely specify the dynamics.
The concept of “full state estimation” is fundamentally different from that of “reduced state estimation.” In the full state estimation context, the state estimation technique attempts to learn the unknown parameters (such turn rate ω and tangential acceleration α in the above mentioned airplane example). In the reduced state estimator, the estimator is not designed to perform any learning at all. In the airplane example, the bounded parameters ω and α are expected to change during the learning process, so that, at any given time, a learned parameter, such as ω or α, is likely to bear no relation to the actual parameter at that time.
According to Portmann, Moore, and Bath (supra), a full state estimator assumes “that accelerations last long enough and are constant enough to be observed and estimated.” Li and Jilkov (supra) observe that a full state estimator “suffers from two major deficiencies, which stem from assuming constant input and known onset time.” Except in the case of target maneuvers, target trajectories are very predictable. Since the onset time of a maneuver is not known, maneuvers are difficult to model as stochastic processes. For this reason, full state estimators are rarely used to track maneuvering targets. Kalman filters with white plant noise are currently used as reduced state estimators. Such Kalman filters are not necessarily optimal. Portmann, Moore and Bath could not solve the problem beyond a single parameter in a one-dimensional tracking scenario. They state in their article that their filter “can be modified in a straightforward manner to permit operation in two or three coupled dimensions” and that “The major differences” in one dimension versus multiple dimensions “lie in the special treatment of the cross-gain terms when computing the lags and in the form of the minimization process.” However, their method cannot be generalized beyond one dimension, and no such solution was ever published by them or by anyone else. In particular, their use of absolute values and signs of their single parameter cannot be generalized (see equations (20) and (21) in G. J. Portmann, J. R. Moore, and W. G. Bath, “Separated Covariance Filtering,” Record of the IEEE 1990 International Radar Conference, 1990, pp. 456–460).
Bar-Shalom, Blair, Li, Moore, and Kirubarajan (supra) define a track filter to be consistent if the state errors and innovations (i.e., measurement residuals) satisfy the tenets of Kalman filter theory, namely that the state estimation and innovation covariances correctly characterize the actual errors, and that the innovations are a white stochastic process as additionally set forth in W. D. Blair and Y. Bar-Shalom, “Tracking Maneuvering Targets with Multiple Sensors: Does More Data Always Mean Better Estimates?” IEEE Transactions on Aerospace and Electronic Systems, pp. 450–456, Vol. AES-32, No. 1, January 1996 and J. R. Moore and W. D. Blair, “Practical Aspects of Multisensor Tracking,” in Multitarget-Multisensor Tracking: Applications and Advances, Volume III, Y. Bar-Shalom and William Dale Blair, (ed.) Boston, Mass.: Artech House, 2000, pp. 43–44. Specifically, these three conditions for Kalman filter consistency are as follows:    (a) “The state errors should be acceptable as zero mean and have magnitude commensurate with the state covariance as yielded by the filter.”    (b) “The innovations should also have the same property.”    (c) “The innovations should be acceptable as white.”Note that a reduced state estimator can only satisfy the first two of these conditions for consistency, since the innovations of a reduced state estimator, unlike a full state estimator, are generally not a white stochastic process. Thus, we define a reduced state estimator to be consistent if the state estimation and innovation covariances correctly characterize the actual errors. That is, a reduced state estimator is consistent if conditions (a) and (b) above are satisfied.
Another definition of filter consistency from Bar-Shalom, Li, and Kirubarajan is that “A state estimator is consistent if the first and second order moments of its estimation errors are as the theory predicts.” This definition also applies to reduced state estimators, and is satisfied if the RMS state estimation errors lie within the one-sigma error ellipsoid of the state covariance as calculated by the filter. As stated by Bar-Shalom, Li, and Kirubarajan, “Since the filter gain is based on the filter-calculated error covariances, it follows that consistency is necessary for filter optimality: Wrong covariances yield wrong gain. That is why consistency evaluation is vital for verifying a filter design—it amounts to evaluation of estimator optimality.”
Consequences of filter inconsistency for tracking of maneuvering targets may be:    (a) Estimated trajectories deviate substantially from the true trajectory during maneuvers;    (b) Covariance matrices produced by the filter do not characterize the actual errors, and hence decision processes that rely on these covariances are subject to unwanted erroneous or false decisions; and    (c) Multisensor tracking yields worse errors than single-sensor tracking.It has been shown in P. Mookerjee and F. Reifler, “Optimal Reduced State Estimators for Consistently Tracking Maneuvers,” IEEE Transactions on Aerospace and Electronic Systems (in press), that the inconsistency causing these adverse consequences can be resolved using the optimal reduced estimator derived by P. Mookerjee and F. Reifler in “Reduced State Estimator for Systems with Parametric Inputs,” IEEE Transactions on Aerospace and Electronic Systems, pp. 446–461, Vol. AES-40, No. 2, April 2004.
According to Moore and Blair (supra), “Track filter consistency is critical for effective fusion of data from multiple sensors with diverse accuracies. Maneuvering targets pose a particularly difficult challenge to achieving track filter consistency.” Blair and Bar-Shalom (supra) have shown an example where a Kalman filter used as an inconsistent reduced state estimator paradoxically yields worse errors with multisensor tracking than with single sensor tracking.
Note that when a filter is used to support a decision process, such as collision avoidance or detection-to-track correlation, the measure of performance is the frequency of false decisions. As stated by Portmann, Moore, and Bath (supra), “At any decision point in time (not necessarily at the time of a measurement), one needs both the best available estimate of object state and a firm confidence interval for this state that allows one to say with specified probability that the object state is in some region about the estimate regardless of whether it is accelerating and regardless of how long it has been accelerating. The confidence interval should be valid for an extreme target acceleration sequence which is based on what is known about target dynamics.”
As a reduced state estimator, the Kalman filter suffers from several difficulties in addition to its inconsistency for supporting decisions as discussed in the above mentioned Portmann, Moore, and Bath references, and in P. Mookerjee and F. Reifler, “Application of Reduced State Estimation to Multisensor Fusion with Out-of-Sequence Measurements,” Proceedings of the 2004 IEEE Radar Conference, Philadelphia, Pa., Apr. 26–29, 2004, pp. 111–116.
FIG. 2 is a simplified logic flow chart or diagram 200 illustrating a reduced-state Kalman filter such as may be found in the prior art. A similar illustration is available in the textbook Y. Bar-Shalom, X. Rong Li, and T. Kirubarajan (2001), Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software, John Wiley & Sons, Inc. New York. FIG. 2 applies the reduced state Kalman filter to the problem of estimating the state of a system described byx(k+1)=Φx(k)+Γu(x(k),λ)  (1)where x(k) is the state vector at the kth sample time tk for k=0, 1, 2, . . . , and u(x(k),λ) is a system input that is a function of the state vector x(k) and arbitrarily unknown time-varying parameters λ with known bounds. This input function u(x(k),λ) may be nonlinear or linear. Here the matrices Φ and Γ are the system transition and input matrices at time tk. In general, these system matrices represent the relationship between the current state and the previous state of the system. The parameters λ are neither constant nor stochastic processes. These parameters have a known mean value λ. (This general class of systems is of type (c) or (e) discussed below in paragraph 0037.) A limitation of the Kalman filter is that only the mean value λ can be used in the reduced state Kalman filter and not their physical bounds. The Kalman filter's only recourse to addressing the physical bounds on the parameters λ is to use an empirically determined white noise covariance W that does not accurately model the physical bounds in the prior art. A sensor (or sensors) measures aspects of the system's state. In the target tracking example, the state of the system might be target position and velocity and a measured aspect of the state might be position alone. Measurements z(k) are collected according toz(k)=Hx(k)+n(k)  (2)where at time tk, the matrix H is the measurement matrix and n(k) is the kth sample of the measurement noise, whose covariance matrix is N.
In FIG. 2, the logic of the prior art Kalman filter begins with a block 210, which initializes the state estimate {circumflex over (x)}(k0|k0) and the state covariance S(k0|k0), where k0=0, 1, 2, . . . measurements are used for initialization at time tk0. In general, the notations {circumflex over (x)}(j|k) and S(j|k) will denote the state estimate and its covariance at time t for j=1, 2, . . . based on processing k measurements at times t1, t2, . . . , tk. From block 210, the logic of the prior art Kalman filter of FIG. 2 flows to a block 212, which represents the incrementing of a time index from k to k+1.
Block 214 of the prior art Kalman filter represents the accessing or inputting of system transition matrices Φ, Γ, F, and G, where
                                          F            =                          Φ              +                              Γ                ⁢                                                      ∂                    u                                                        ∂                    x                                                                                                                      x            =                                          x                ^                            ⁡                              (                                  k                  ❘                  k                                )                                              ,                      λ            =                          λ              _                                                          (        3        )                                                      G            =                          Γ              ⁢                                                ∂                  u                                                  ∂                  λ                                                                                                    x            =                                          x                ^                            ⁡                              (                                  k                  ❘                  k                                )                                              ,                      λ            =                          λ              _                                                          (        4        )            From block 214, the logic of the prior art Kalman filter of FIG. 2 flows to a block 216, which represents the accessing or inputting of the white process noise covariance W. The white process noise covariance W is determined in an empirical way in the prior art.
From block 216, the logic of the prior art Kalman filter of FIG. 2 proceeds to a block 218, which represents the extrapolation of state estimate and covariance. The extrapolated state is{circumflex over (x)}(k+1|k)=Φ{circumflex over (x)}(k|k)+Γu({circumflex over (x)}(k|k),λ)  (5)and its covariance isS(k+1|k)=FS(k|k)F′+GWG′  (6)From block 218, the logic of the prior art Kalman filter of FIG. 2 flows to a block 220, which represents the accessing or inputting of the measurement noise covariance N. Block 222 represents the computation of the filter gain matrix K. The covariance, Q, of the residual is calculated asQ=HS(k+1|k)H′+N  (7)The filter gain matrix K is calculated asK=S(k+1|k)H′Q−1  (8)and the matrix L is calculated asL=I−KH  (9)where I is the identity matrix.
From block 222 of FIG. 2, the logic of the prior art Kalman filter flows to a block 224, which represents the accessing or inputting of sensor measurements z(k+1). Finally, block 226 of FIG. 2 represents updating the state estimate and its covariance given by{circumflex over (x)}(k+1|k+1)={circumflex over (x)}(k+1|k)+K[z(k+1)−H{circumflex over (x)}(k+1|k)]  (10)S(k+1|k+1)=LS(k+1|k)L′+KNK′  (11)Prior art (Kalman filter) uses the white process noise covariance W in (6) and obtains the optimal gain matrix K that minimizes the updated state covariance S(k+1|k+1) in (11).
Improved or alternative state estimation is desired.